I think that would be translated to
"De kaas is best (wel) oud"
And when you translate that back again you would likely say
"The cheese is pretty old"
So quite indicates a bigger amount than a little bit. Plus if a sentences isn't translated as closely as possible there is room for variation and interpretation.
I guess in this case you could blame the use of "a bit" for it, because sometimes it is used to mean an actually small amount and sometimes indeed it is used like quite/rather.